Link to actual problem [8962] \[ \boxed {y^{\prime }-\frac {x \left (-2+3 \sqrt {x^{2}+3 y}\right )}{3}=0} \]
type detected by program
{"first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[[_1st_order, `_with_symmetry_[F(x),G(y)]`]]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\).\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= \frac {3 x}{2}, \underline {\hspace {1.25 ex}}\eta &= x^{2}+6 y\right ] \\ \left [R &= \frac {x^{2}+3 y}{3 x^{4}}, S \left (R \right ) &= \frac {2 \ln \left (x \right )}{3}\right ] \\ \end{align*}
My program’s symgen result This shows my program’s found \(\xi ,\eta \) and the corresponding ODE in canonical coordinates \(R,S\).\begin{align*} \xi &= 0 \\ \eta &=-\frac {3 x^{2} \sqrt {x^{2}+3 y}}{2}+2 x^{2}+6 y \\ \frac {dS}{dR} &= 0 \\ \end{align*}